Math
Vector Addition Calculator
Type any number of vectors, one per line, and mathcheck adds them (or subtracts the rest from the first) component by component.
It returns the resultant vector, its exact magnitude and direction angle, and draws the tip-to-tail diagram, including the parallelogram for two vectors.
Separate components with commas. In 3D mode each line needs three numbers.
Try an example
Result
Resultant vector
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- Magnitude of resultant
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- Direction of resultant
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- Expression
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
How vector addition works
A quantity with both a size and a direction, often drawn as an arrow. add component by component: the x-components add to give the x-component of the sum, and the same for y (and z). Geometrically this is the tip-to-tail rule: draw the second vector starting where the first one ends, and the resultant runs from the start of the first to the tip of the last. For two vectors, the resultant is the diagonal of the parallelogram they form.
⟨3, 4⟩ + ⟨-1, 2⟩
x: 3 + (-1) = 2. y: 4 + 2 = 6. Resultant ⟨2, 6⟩ with magnitude sqrt(4 + 36) = 2√10 ≈ 6.32 and direction atan2(6, 2) ≈ 71.57°.
Adding vectors given by magnitude and angle
If a vector is described as a length r at angle θ, convert it to components first: x = r cos θ and y = r sin θ. Add the components, then convert the resultant back to The length or size of a vector, ignoring which way it points. and direction. Adding the magnitudes directly is only correct when the vectors point the same way.
How to use this calculator
- Choose 2D or 3D and whether to add or subtract.
- Type each vector on its own line, components separated by commas.
- Read the resultant in the header with its magnitude beside it.
- Check the diagram: each arrow starts where the previous one ends, and the red arrow is the resultant.
- Open Show the work for the component-by-component arithmetic.
How to read the answer
The resultant is the single vector that has the same effect as all the vectors combined, such as the net force or net displacement. Its magnitude is usually less than the sum of the individual magnitudes unless all vectors point the same way. A zero resultant means the vectors cancel, which is the condition for equilibrium in physics.
Common mistakes and edge cases
- Adding magnitudes instead of components. |u + v| equals |u| + |v| only when u and v point in the same direction.
- Subtracting in the wrong order. u - v points from the tip of v to the tip of u, and v - u is the opposite.
- Mixing 2D and 3D vectors in one list; every vector needs the same number of components.
- Forgetting to convert magnitude-and-angle form into components before adding.
- Reading the direction of a resultant in One of the four regions the x- and y-axes cut the coordinate plane into. II or III with a plain arctangent; the calculator uses atan2.
Worked examples
Two 2D vectors
⟨3, 4⟩ + ⟨-1, 2⟩
Resultant vector
⟨2, 6⟩
Subtraction
⟨5, 1⟩ - ⟨2, 4⟩
Resultant vector
⟨3, -3⟩
Three vectors
⟨1, 0⟩ + ⟨0, 1⟩ + ⟨2, 2⟩
Resultant vector
⟨3, 3⟩
3D vectors
⟨1, 2, 3⟩ + ⟨4, -5, 6⟩
Resultant vector
⟨5, -3, 9⟩
Vectors that cancel
⟨2, -3⟩ + ⟨-2, 3⟩
Resultant vector
⟨0, 0⟩
Decimal components
⟨1.5, -0.5⟩ + ⟨0.25, 2⟩
Resultant vector
⟨1.75, 1.5⟩
Wrong dimension
A 3-component vector in 2D mode
Resultant vector
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Frequently asked questions
How do you add two vectors?+
Add their matching components: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩. Geometrically, place the tail of the second vector at the tip of the first; the sum runs from the first tail to the last tip.
What is the resultant vector?+
The single vector equal to the sum of all the vectors. In physics it is the net force or net displacement; its magnitude and direction come from the summed components.
What is the parallelogram law?+
When two vectors start from the same point, they form two sides of a parallelogram, and their sum is the diagonal from that common point. It gives the same answer as the tip-to-tail rule.
How do I subtract vectors?+
Subtract matching components, or add the opposite vector: u - v = u + (-v). The difference points from the tip of v to the tip of u.
Can I add vectors given as magnitude and angle?+
Convert each to components first with x = r cos θ and y = r sin θ, add the components, then convert the resultant back with the magnitude and direction formulas.
Why is the magnitude of the sum less than the sum of the magnitudes?+
Because the vectors point in different directions and partly cancel. The triangle inequality says |u + v| ≤ |u| + |v|, with equality only when u and v point the same way.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026